{
  "id": "the-cartier-miller-evaluation",
  "title": "The Cartier–Miller evaluation of coefficient sums",
  "description": "Truncated powers of a polynomial of degree three or four over a prime field and the curve v^2 = f(w): the coefficient that gives the trace of the curve, the truncated sum as a sum over the points of the curve and as a character sum, the Cartier operator on the function field of the curve with its values on the two differentials that carry the sums, the points of the curve with orders and residues, functions and differentials without poles, the regular identity that evaluates the sums through a function on the curve, and the statement of D. Jordan's theorem (2026) that evaluates these sums with about log p operations through the arithmetic of the curve.",
  "units": [
    {
      "id": "truncated-powers-and-points-of-a-curve",
      "title": "Truncated powers and points of a curve",
      "reader": "truncated-powers-and-points-of-a-curve.html",
      "source": "src/truncated-powers-and-points-of-a-curve.md",
      "source_sha256": "2846736e33f738fd66d8e9c8f2595705c4a43aebc4fb14430232e40468481311",
      "status": "author_self_checked"
    },
    {
      "id": "the-cartier-operator",
      "title": "The Cartier operator",
      "reader": "the-cartier-operator.html",
      "source": "src/the-cartier-operator.md",
      "source_sha256": "2cfc475b7846129dde9ce38522123019e7d736dc551ce5ff3261a41e17453be2",
      "status": "author_self_checked"
    },
    {
      "id": "points-orders-and-residues",
      "title": "Points, orders and residues",
      "reader": "points-orders-and-residues.html",
      "source": "src/points-orders-and-residues.md",
      "source_sha256": "d6a30b3ec2509763cb3b5c36b53d6a9c08c3c5fc5d0c6664bf6454d1786da287",
      "status": "author_self_checked"
    },
    {
      "id": "the-regular-identity",
      "title": "The regular identity",
      "reader": "the-regular-identity.html",
      "source": "src/the-regular-identity.md",
      "source_sha256": "e3c6d7c3db53564389df300d1ebb4e0583a7b26dda65a59ad1f933a1d046675d",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "core course Abstract Algebra II (C40): finite fields, polynomials, the number of roots of a polynomial",
    "the second lesson uses derivations and formal power series (core course Abstract Algebra II)",
    "the first lesson proves every other fact it uses"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}